Claim 2 there is no integer k such that k is odd and k2 is even. Notice that we did not alert the reader that we are doing a proof of the contrapositive. This video is part of a discrete math course taught at the university of cinc. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. The contrapositive is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. Assuming the logic is sound, the only option is that the assumption that p is not true is incorrect. Contrapositive proof example proposition suppose n 2z. The reason is that the proof setup involves assuming x,px, which as we know from section 2. Conclusion that proof of the contrapositive proves the original statement example if a sum of two real numbers is less than 50, then at least one of the numbers is less than 25.
Copious examples of proofs 19 rewrite it in each of the three forms and prove each. Similarly, a statements converse and its inverse are always either both true or both false. The method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive. Therefore, there is a such that, is prime and is even, all at the same time since is even, we can write for some now, since, as well, cannot be equal to as a result, can be written as the product of and another number that is not. Example 9 write the contrapositive of the following statement. The contrapositive of a conditional statement always has the same truth value as the original statement. From the truth table, the columns of conditional statement and contrapositive statement are same. Hence, contrapositive of a statement is described as reversal and negation both the statements. The method of contradiction is an example of an indirect proof. Mat231 transition to higher math direct proof fall 2014 14 24. If 15 is not a prime number, then 15 is not an odd number.
It is very important to realize that these two implications are not logically equivalent. To prove that p q is false, find an example of where p is true. Prove the following statement by contraposition if a product of two positive real numbers is greater than 100, then at. Suppose we have a set, s, and that t is a subset of s, as shown in the diagram below. Unlike the method of trivial proof, this method may be applied to any implication. However, we can still write down what a counterexample must look like. Proof by contraposition examples kent state university. Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements.
It just means that youre referring to the only other way in the lawgic language of expressing that particular conditional relationship. Proof by contradiction a proof by contradiction is a proof that works as follows. Contrapositive example with sets the infinite series module. What are the converse, contrapositive, and inverse. There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. For our next example, consider the following proposition concerning an integer x. From this assumption, p 2 can be written in terms of a b, where a and b have no common factor. Whenever a conditional statement is true, its contrapositive is also true and vice versa. The empty set is a subset of a, hence it is an element of the power set of a. Existence proof examples show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways.
For example, the assertion if it is my car, then it is red is equivalent to if that car is not red, then it is not mine. You are probably familiar with at least a little canadian geography, and know that british columbia bc is a geographical area within canada. Example 9 write contrapositive of i if a number examples. Contrapositive definition of contrapositive by merriam. Pdf a statement, the contrapositive and the inverse. If three angles are equal in a triangle, then the triangle is an equilateral.
So this is a valuable technique which you should use sparingly. Based on the assumption that p is not true, conclude something impossible. Contrapositive definition is a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition or theorem and interchanging them. Basic proof techniques washington university in st. The basic concept is that proof by contrapositive relies on the fact that p. We now introduce a third method of proof, called proof by contra. The statement \a implies b can be written symbolically. In mathematic logic, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive. I will assume that x is odd and y is even without loss of generality, since x and y are commutative. Logical equivalence guarantees that this is a valid proof method.
The negation of a statement simply involves the insertion of the word not at the proper part of the statement. The colors show how the numbers move from one line to the next based on the lemma we just proved. We present our intended understanding of contrapositive equivalence in terms of simons 2017 explication of mathematical concepts. D s is the contrapositive of s d just as much as s d is the contrapositive of d s. The proves the contrapositive of the original proposition. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. If an element y is in t, then y must also be in s, because t, is a subset of s. From rstorder logic we know that the implication p q is equivalent to. While weve seen that its possible for a statement to be true while its converse is false, it turns out that the contrapositive is better behaved. Bc is a province belonging to canada, as shown in the diagram.
The contrapositive of the statement has its antecedent and consequent inverted and flipped. P is true, and often that is enough to produce a contradiction. So, to prove if p, then q by the method of contrapositive means to prove if not q, then not p. The underlying reasoning is that since a conditional statement is logically equivalent to its contrapositive, if the contrapositive is true, then the statement must also be true. Logic and proof converse, inverse, and contrapositive. Our next example follows their logic to prove that 2 is. Parity here is a simple example that illustrates the method.
Chapter 6 proof by contradiction mcgill university. For our next example, consider the following proposition concerninganintegerx. Therefore, the contrapositive of a definition is always true. For any set a, the empty set is an element of the power set of a. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs. Notes on proof by contrapositive and proof by contradiction. In other words, the conclusion if a, then b is inferred by constructing a proof of the claim if not. If a number is not divisible by 3, it is not divisible by 9 ii if you are born in india, then you are a citizen of india. The second proposition is called the contrapositive of the rst proposition.
To prove that p is true, assume that p is not true. Shorser the contrapositive of the statement \a b i. In a direct proof you start with the hypothesis of a statement. Proof by contradiction often works well in proving statements of the form. By the closure property, we know b is an integer, so we see that 3jn2. Let us assume that the original statement is false. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. Since q2 is an integer and p2 2q2, we have that p2 is even. The symbols for contrapositive may be read as not q implies not p or if not q, then not p. Proofs by contradiction computer science university of.
Intro to proofs proof by contrapositive involving an. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. So, to prove if p, then q by the method of contrapositive means to prove if. Though the proofs are of equal length, you may feel that the contrapositive proof. Euclidean algorithm to nd the gcd lets use the euclidean algorithm to nd gcd38. Converses, contrapositives and proof by the contrapositive. Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. The hypothesis is false, therefore the statement is vacuously true even though the conclusion is also false. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. By denition, the reverse of an implication means the same as the original implicationitself. The three forms are 1 direct if n2 is even, n is even. Contrapositive examples the infinite series module.
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